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 A034705 Numbers that are sums of consecutive squares. 31
 0, 1, 4, 5, 9, 13, 14, 16, 25, 29, 30, 36, 41, 49, 50, 54, 55, 61, 64, 77, 81, 85, 86, 90, 91, 100, 110, 113, 121, 126, 135, 139, 140, 144, 145, 149, 169, 174, 181, 190, 194, 196, 199, 203, 204, 221, 225, 230, 245, 255, 256, 265, 271, 280, 284, 285, 289, 294, 302 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also, differences of any pair of square pyramidal numbers (A000330). These could be called "truncated square pyramidal numbers". - Franklin T. Adams-Watters, Nov 29 2006 If n is the sum of d consecutive squares up to m^2, n = A000330(m) - A000330(m-d) = d*(m^2 - (d-1)m + (d-1)(2d-1)/6 <=> m^2 - (d-1)m = c := n/d - (d-1)(2d-1)/6 <=> m = (d-1)/2 + sqrt((d-1)^2/4 + c) which must be an integer. Moreover, A000330(x) >= x^3/3, so m and d can't be larger than (3n)^(1/3). - M. F. Hasler, Jan 02 2024 LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 Index entries for sequences related to sums of squares EXAMPLE All squares (A000290: 0, 1, 4, 9, ...) are in this sequence, since "consecutive" in the definition means a subsequence without interruption, so a single term qualifies. 5 = 1^2 + 2^2 = A000330(2) is in this sequence, and similarly 13 = 2^2 + p3^2 = A000330(3) - A000330(1) and 14 = 1^2 + 2^2 + 3^2 = A000330(3), etc. MATHEMATICA nMax = 1000; t = {0}; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, AppendTo[t, s]; k++], {n, Sqrt[nMax]}]; t = Union[t] (* T. D. Noe, Oct 23 2012 *) PROG (Haskell) import Data.Set (deleteFindMin, union, fromList); import Data.List (inits) a034705 n = a034705_list !! (n-1) a034705_list = f 0 (tail \$ inits \$ a000290_list) (fromList [0]) where f x vss'@(vs:vss) s | y < x = y : f x vss' s' | otherwise = f w vss (union s \$ fromList \$ scanl1 (+) ws) where ws@(w:_) = reverse vs (y, s') = deleteFindMin s -- Reinhard Zumkeller, May 12 2015 (PARI) {is_A034705(n)= for(d=1, sqrtnint(n*3, 3), my(b = (d-1)/2, s = n/d - (d-1)*(d*2-1)/6 + b^2); denominator(s)==denominator(b)^2 && issquare(s, &s) && return(b+s)); !n} \\ Return the index of the largest square of the sum (or 1 for n = 0) if n is in the sequence, else 0. - M. F. Hasler, Jan 02 2024 (Python) import heapq from itertools import islice def agen(): # generator of terms m = 0; h = [(m, 0, 0)]; nextcount = 1; v1 = None while True: (v, s, l) = heapq.heappop(h) if v != v1: yield v; v1 = v if v >= m: m += nextcount*nextcount heapq.heappush(h, (m, 1, nextcount)) nextcount += 1 v -= s*s; s += 1; l += 1; v += l*l heapq.heappush(h, (v, s, l)) print(list(islice(agen(), 60))) # Michael S. Branicky, Jan 06 2024 CROSSREFS Cf. A000290, A000330, A034706. Cf. A217843-A217850 (sums of consecutive powers 3 to 10). Cf. A368570 (first of each pair of consecutive integers in this sequence). Sequence in context: A060199 A229240 A322135 * A006844 A022425 A277549 Adjacent sequences: A034702 A034703 A034704 * A034706 A034707 A034708 KEYWORD nonn AUTHOR Erich Friedman EXTENSIONS Terms a(1..10^4) double-checked with independent code by M. F. Hasler, Jan 02 2024 STATUS approved

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Last modified March 4 11:02 EST 2024. Contains 370528 sequences. (Running on oeis4.)